Upper Bound for Lucas Number
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Theorem
Let $L_n$ denote the $n$th Lucas number.
Then:
- $L_n < \paren {\dfrac 7 4}^n$
Proof
The proof proceeds by complete induction.
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the proposition:
- $L_n < \paren {\dfrac 7 4}^n$
$\map P 1$ is the case:
\(\ds L_1\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \dfrac 7 4\) |
Thus $\map P 1$ is seen to hold.
Basis for the Induction
$\map P 2$ is the case:
\(\ds L_2\) | \(=\) | \(\ds 3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {48} {16}\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \dfrac {49} {16}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\dfrac 7 4}^2\) |
Thus $\map P 2$ is seen to hold.
This is the basis for the induction.
Induction Hypothesis
Now it needs to be shown that if $\map P j$ is true, for all $j$ such that $0 \le j \le k$, then it logically follows that $\map P {k + 1}$ is true.
This is the induction hypothesis:
- $L_k < \paren {\dfrac 7 4}^k$
from which it is to be shown that:
- $L_{k + 1} < \paren {\dfrac 7 4}^{k + 1}$
Induction Step
This is the induction step:
\(\ds L_{k + 1}\) | \(=\) | \(\ds L_k + L_{k - 1}\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \paren {\dfrac 7 4}^k + \paren {\dfrac 7 4}^{k - 1}\) | Induction Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\dfrac 7 4}^{k - 1} \paren {1 + \dfrac 7 4}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\dfrac 7 4}^{k - 1} \paren {\dfrac {11} 4}\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \paren {\dfrac 7 4}^{k - 1} \paren {\dfrac 7 4}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\dfrac 7 4}^{k + 1}\) |
So $\map P k \implies \map P {k + 1}$ and the result follows by the Second Principle of Mathematical Induction.
Therefore:
- $\forall n \in \Z_{\ge 1}: L_n < \paren {\dfrac 7 4}^n$
$\blacksquare$
Sources
- 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction: Example $1 \text{-} 1$